Properties

Label 303450.b
Number of curves $6$
Conductor $303450$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 303450.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
303450.b1 303450b6 \([1, 1, 0, -12590140650, -543747958852500]\) \(585196747116290735872321/836876053125000\) \(315627398074255517578125000\) \([2]\) \(509607936\) \(4.3560\)  
303450.b2 303450b3 \([1, 1, 0, -1825179650, 30007268392500]\) \(1782900110862842086081/328139630024640\) \(123757702521159682815000000\) \([2]\) \(254803968\) \(4.0094\)  
303450.b3 303450b4 \([1, 1, 0, -794027650, -8334185895500]\) \(146796951366228945601/5397929064360000\) \(2035826332001483450625000000\) \([2, 2]\) \(254803968\) \(4.0094\)  
303450.b4 303450b2 \([1, 1, 0, -125859650, 366029632500]\) \(584614687782041281/184812061593600\) \(69701779511683905600000000\) \([2, 2]\) \(127401984\) \(3.6628\)  
303450.b5 303450b1 \([1, 1, 0, 22108350, 38872384500]\) \(3168685387909439/3563732336640\) \(-1344059924580925440000000\) \([2]\) \(63700992\) \(3.3163\) \(\Gamma_0(N)\)-optimal
303450.b6 303450b5 \([1, 1, 0, 311397350, -29720843370500]\) \(8854313460877886399/1016927675429790600\) \(-383533780214002738485178125000\) \([2]\) \(509607936\) \(4.3560\)  

Rank

sage: E.rank()
 

The elliptic curves in class 303450.b have rank \(1\).

Complex multiplication

The elliptic curves in class 303450.b do not have complex multiplication.

Modular form 303450.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - 4q^{11} - q^{12} - 6q^{13} + q^{14} + q^{16} - q^{18} + 4q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.